Question

Solve the following system of equations and show all work:

y = -x^2 + 4
y = 2x + 1
A) x = 2, y = 5
B) x = -2, y = -3
C) x = 1, y = 3
D) x = -1, y = 1

Answer 1

To solve the system of equations, set the two equations equal to each other and solve for x. Then substitute the values of x into either equation to solve for y. The solutions are (x = 1/3, y = 35/9) and (x = -3, y = -5).

Step-by-step explanation:

To solve the system of equations:

y = -x² + 4

y = 2x + 1

1. Set the two equations equal to each other:
   -x² + 4 = 2x + 1
2. Rearrange the equation:
   3x^2 + 2x - 3 = 0
3. Factor the quadratic equation:
   (3x - 1)(x + 3) = 0
4. Set each factor equal to zero and solve for x:
   3x - 1 = 0 or x + 3 = 0
5. For the first equation:
   3x - 1 = 0
   3x = 1
   x = 1/3
6. For the second equation:
   x + 3 = 0
   x = -3
7. Substitute the values of x into either equation to solve for y:
   For x = 1/3:
   y = - (1/3)² + 4
   y = -1/9 + 4
   y = 35/9
8. For x = -3:
   y = -(-3)²+ 4
   y = -9 + 4
   y = -5

Therefore, the solutions to the system of equations are (x = 1/3, y = 35/9) and (x = -3, y = -5).